User:Joeblakesley
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Unterstand a1 to be the arithmetic mean of x and y, i.e. (x+y)/2.
g1 is their geometric mean, i.e. the square root of xy.
If we define the sequences {a} and {g} such that:
- an+1=(an+gn)/2
and
- gn+1=(angn)1/2
then both of these sequences converge to the same number, which is the arithmetic-geometric mean of x and y.
The geometric harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. The arithmetic harmonic mean is none other than the geometric mean.
If M(x,y) is the arithmetic geometric mean of x and y, and K is the elliptic integral of the first kind, then
M(x,y)=π(x+y)/4K((x-y)/(x+y))